Special Relativity: Tensor Calculus and Four-Vectors
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Special Relativity: Tensor Calculus and Four-Vectors Looking ahead to general relativity, where such things are more important, we will now introduce the mathematics of tensors and four-vectors. The Mathematics of Spacetime Let's start by de¯ning some geometric objects. Bear with me for the ¯rst couple, which seem obvious but lay the groundwork for the less obvious sequels. Scalar.|A scalar is a pure number, meaning that all observers will agree on its value. For example, the number \3" is a scalar. Okay, that one's trivial, but there are others that aren't so obvious, and we'll get to those in a bit. Event.|Next we have an event. An event is e®ectively a \point" in spacetime. More generally, if you have an N-dimensional space, you need N numbers to label it uniquely. For example, in two dimensions you need two numbers; e.g., x and y for a plane, or and Á for the surface of a sphere. For spacetime, you need four numbers: e.g., t, x, y, and z. Naturally, the essence of the event isn't changed if you relabel the coordinates. I want to stress this, because something that is obvious for events but may not be obvious for some other geometric objects is that although when you ¯nally calculate something you may choose a coordinate system and break things into components, there is also an independent reality (well, within the math at least!) of the objects. Going with the coordinate-free representation has proved very helpful in proving theorems about spacetime, but when doing astrophysics it is usually best to investigate components in some given system. Vector.|Next, consider a vector. Normally we think about three-dimensional vectors, but here we have to consider four-vectors. This is something that has four numbers; for example, the location of an event can be written as x¹ = (ct; x; y; z). You can also imagine a line segment drawn between two events: this is a vector as well. Four-velocities are also vectors. In the geometric sense, it is important to work with four-vectors rather than the more familiar purely spatial three-vectors. The reason is just as an event can be thought of as a purely geometric object with existence independent of the coordinate system, so can a four- vector. This is not the case with three-vectors. For example, consider an electric ¯eld at a given location. This is a three-vector. You know from electromagnetism that under many circumstances you can boost into a frame where the electric ¯eld vanishes. Three-vectors, then, are not coordinate-independent geometric objects. One-form.|Events and vectors are pretty familiar. Not so with the next object. This is a one-form. Think again about a Euclidean plane, and two points very close to each Fig. 1.| Vectors and their corresponding one-forms. From http://gregegan.customer.netspace.net.au/FOUNDATIONS/03/Fig04.gif other. You can de¯ne a vector between them. But you could equally well de¯ne something perpendicular to that vector (see the ¯gure). If you imagined two points in three dimensions, then after drawing a vector you could draw a plane perpendicular to it. In four dimensions (spacetime), you would have a three-dimensional thing perpendicular to the vector. This is called a one-form. One-forms are written di®erently than vectors. For example, consider the radial component of a velocity v. It is written vr, with a superscript. The r-component of the corresponding one-form would be written vr, with a subscript. These are also called, respectively, contravariant (up) and covariant (down) components. In the GR lectures will get into how to transform between the two, by raising and lowering indicies. Tensor.|One can generalize with the further concept of tensors. Think, for example, of the gradient of an electric ¯eld: K ´ rE (this isn't a tensor itself, because it involves just the spatial components, but we're using this as an analogy). At a given point, if you want to know the components of this you can't just specify its x or y component, for example. Instead, you need to specify things like the gradient of the x component of E, in the y direction. Then xy K might have components like K or Kzz, depending on whether one wanted to go with a contravariant or covariant description. We'll get to how to manipulate tensors and their indices a little later. For now, it is also useful to think of a tensor as a machine that can take vectors or one-forms as inputs in its \slots", one slot per index, and return a number. It is a linear machine. As with vectors and one-forms, a tensor has a mathematical existence of its own, independent of coordinate systems, but when calculating it is usually convenient to select a particular coordinate system and compute with components. The \rank" of the tensor is the number of separate indices it has. For example, T ¹º is a ® second-rank tensor and R¯± is a fourth-rank tensor. One especially important second-rank tensor is the metric tensor, which we'll talk about now. Metrics Now let's move a little from those basic de¯nitions to how they are used in curved spacetime (yes, we're dealing with at spacetime now but we want an easy generalization to other spacetimes). A great way to characterize the curvature is through the use of a metric. This e®ectively tells you the \distance" between two events. For example, in two dimensions, what is the distance between two events separated by dx in the x direction and dy in the y direction? The distance ds is just given by ds2 = dx2 + dy2. In polar coordinates we would write ds2 = dr2 + r2d2, but it's the same thing. One point (obvious here, not so obvious later) is that this distance is the same for two given events even if the coordinates are rede¯ned (e.g., by rotation of the coordinate system). Therefore, as we did in the last lecture, we call this the invariant interval between the events. In four dimensions and at space, special relativity tells us that the invariant interval is de¯ned as ds2 = ¡c2dt2 + dx2 + dy2 + dz2 : (1) For two arbitrary events, ds2 can be positive, negative, or zero. Ask class: consider just dt and dx, so that ds2 = ¡c2dt2 + dx2. What is the condition that ds2 = 0? This is the condition that the two events could be connected by a photon going from one to the other. Therefore, ds2 = 0 is the path of a light ray. We can represent this more compactly, using the metric tensor g®¯, as 2 ® ¯ ds = g®¯dx dx : (2) Here we have introduced two conventions. The ¯rst is the use of greek indices to represent indices that might be any of the four in spacetime (for example, t, x, y, and z). The second is the Einstein summation convention. Whenever you see a symbol used as an up and a down index in the same expression, you are supposed to sum over the four possibilities. For ® t x y z example, v u® = v ut +v ux +v uy +v uz. This also means that if you sum over indices, you ® ®¯ don't count them in the rank of the tensor. For example, v u® is a scalar, T u® is a rank ® 2 one tensor, and R¯ is a rank two tensor. This means that ds is a scalar, so for example it transforms as a scalar does under Lorentz transformations, i.e., it's unchanged (that's why it is an \invariant" interval!). Let's take a moment to emphasize that point a bit more. Scalars are unchanged under Lorentz transformations! At a given \point" (i.e., event) in spacetime, everyone will agree on the value of a scalar. If this isn't clear, just think about a trivial example of a scalar: a pure number. For example, suppose a box contains 3 particles. The number 3 is a scalar. Obviously everyone will agree that the number of the count is 3: countest thou not to 2 unless thou countest also to 3; 5 is right out; and so on. Ask class: what are other examples of scalars? Since scalars are invariant, it often gives insight to construct scalars out of the geometrical quantities of interest. Of course, the particular indices used are dummy indices; we could as well have written 2 ¹ º ¹ º ® ds = g¹ºdx dx . Therefore, dx (or dx or dx or whatever, it's the same thing) is the ¹th component of the separation between the events, in this particular coordinate system. The metric tensor is symmetric: g®¯ = g¯®. Going back to the particular metric we wrote earlier: g®¯ = (¡1; 1; 1; 1) down the diag- onal. This is called the Minkowski metric, and is usually given a special symbol: ´®¯. The Minkowski metric is of special importance. It describes at spacetime, which is spacetime without gravity. Any metric that can be put in the Minkowski form by a transformation also describes at spacetime. Here, by the way, is a place to point out the di®erence be- tween spacetime and a particular metric used to describe it. Spacetime has some particular geometric characteristics (for example, it's at). A metric is what you get when you pick a coordinate system to use with that spacetime. Seems trivial, but as we'll see in a later lec- ture, sometimes a change in the coordinates (which therefore does not change the spacetime) has made a big di®erence in how things are perceived.